Transition matrix and generalized matrix exponential via the Peano-Baker series

We give a closed form for the unique solution to the n x n regressive time varying linear dynamic system of the form x(Delta)(t) = A(t)x(t), x(t(0)) = x(0), via use of a newly developed generalized form of the Peano-Baker series. We develop a power series representation for the generalized time scale matrix exponential when the matrix A(t) equivalent to A is a constant matrix. We also introduce a finite series representation of the matrix exponential using the Laplace transform for time scales, as well as a theorem which allows us to write the matrix exponential as a series of (n - 1) terms of scalar C-rd(infinity)(T, R) functions multiplied by powers of the system matrix A.